Step (REF ) uses many proof arguments from [1]} but we take extra care in keeping track of the rates of various bounds. Proposition REF  of the supplementary materials concludes the discussions of Step (REF ). By our underlying real valued Hilbert space structure, we can apply Riesz's representation theorem to uniquely identify \({H}\)  with its dual \({H}^*\) . Thus we can view the indexing of the supremum of \(\mathcal {R}_n\)  by \(\mathcal {J}_n\)  in Step (REF ) as equivalent to indexing by its dual \(\mathcal {J}_n^*\) , which allows us to apply the tools from empirical process theory. Our desired result for Step (REF ) is the contents of Proposition REF  in the supplementary materials, which is an application of [2]}.
